372 



On the Position of Frames ill a Sloping Gallery. 



[Oct. 



tion. Again let A B, be 

 the direction of the 

 mine carried in as far r~ 

 as B— A C the sur- 



S. 



face of the earth. It 

 is evident that if the 

 roof of the mine was un- 

 supported, the portion 

 A C, would fail in. Let 

 D be the centre of gra- 

 vity of this mass, which 

 is conceived to act as a 



solid body, as the roof- ) 



ing along the top of the 

 mine brings it exactly 

 into the situation of a 

 box of sand. D E will 



then be the direction of the force of gravity ; now it is very evident 

 that a prop, placed upright in the line DE, would bear all the weight of 

 the mass, and that cannot therefore be the best position, but if the prop 

 is placed in the direction DF, the force DE becomes resolved into two 

 others, D F and D G, parallel to the direction of the mine, for as the 

 mass cannot now move in the direction D F, it can only move at right 

 angles to it. Produce DGtoti and make D a equal to the force exerted 

 in that direction, also in D F, take D /3 equal to the force of gravity 

 make ft 8 parallel to D a and equal to it, and complete the parallelo- 

 gram of the resolution of forces, then will D 8 represent the force 

 exerted in the direction D F of the prop, and which from an inspection 

 of the diagram, will be a minimum when the angle D 8 {3 is a right 

 angle; the force D a decreasing as the angle 8 D & decreases, and 

 increasing with its increase, while D 8 increases in both cases, D 

 always remaining constant. The pressure on the prop will then be 

 least, when perpendicular to the direction of the mine, and the weight 

 supported by it, decreasing as the cosine of the angle FDE=CAB= 

 ihe angle of the depression of the mine. 



27 th September, 1838. 



Yours obediently, 



